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</ul>
<h1 class="mume-header" id="%E6%BB%A4%E6%B3%A2%E5%99%A8%E8%AE%BE%E8%AE%A1toc"><a href="#toc">&#x6EE4;&#x6CE2;&#x5668;&#x8BBE;&#x8BA1;</a></h1>

<h2 class="mume-header" id="%E6%A8%A1%E6%8B%9F%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8%E8%AE%BE%E8%AE%A1toc"><a href="#toc">&#x6A21;&#x62DF;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x8BBE;&#x8BA1;</a></h2>

<p>&#x8BB0;&#x6709;&#x6EE4;&#x6CE2;&#x5668;<span class="mathjax-exps">$H_a(j\omega)$</span>, &#x8861;&#x91CF;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x6027;&#x80FD;&#x53C2;&#x6570;&#x4E00;&#x822C;&#x7531;&#x901A;&#x5E26;&#x8FB9;&#x7F18;&#x9891;&#x7387;<span class="mathjax-exps">$\omega_p$</span>, &#x963B;&#x5E26;&#x8D77;&#x59CB;&#x9891;&#x7387;<span class="mathjax-exps">$\omega_s$</span>, &#x901A;&#x5E26;&#x6CE2;&#x7EB9;<span class="mathjax-exps">$\delta_p$</span>(<span class="mathjax-exps">$\alpha_p$</span>), &#x963B;&#x5E26;&#x6CE2;&#x7EB9;<span class="mathjax-exps">$\delta_s$</span>(<span class="mathjax-exps">$\alpha_s$</span>), &#x9009;&#x62E9;&#x53C2;&#x6570;<span class="mathjax-exps">$k$</span>, &#x8FA8;&#x522B;&#x53C2;&#x6570;<span class="mathjax-exps">$k_1$</span>&#x6765;&#x8861;&#x91CF;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; 1 - \delta_p \le |H_a(j\omega)| \le 1 + \delta_p, |\omega| \le \omega_p \\ &amp; |H_a(j\omega)| \le \delta_s, |\omega_s| \le |\omega| \le \infty \\ &amp; \alpha_p = -20 \log_{10}(1-\delta_p) \mathrm{dB} \\ &amp; \alpha_s = -20 \log_{10}(\delta_s) \mathrm{dB} \\ &amp; k = \frac{\omega_p}{\omega_s} \\ &amp; k_1 = \frac{\epsilon}{\sqrt{A^2 - 1}}, A = \frac{1}{\delta_s}, \epsilon = \sqrt{\frac{2\delta_p}{1-\delta_p}} \end{aligned}$$</div><p></p>
<h3 class="mume-header" id="butterworth%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">Butterworth&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;</a></h3>

<p>Butterworth&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x8F6C;&#x79FB;&#x51FD;&#x6570;&#x548C;&#x5E45;&#x9891;&#x54CD;&#x5E94;&#x51FD;&#x6570;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H_a(s) = \frac{z(s)}{p(s)} = \frac{\omega_c^N}{\prod_{l=1}^{N}(s-p_l)}, p_l=\omega_c e^{j[\pi(N+2l-1)/2N]} \\ &amp; |H_a(j\omega)|^2 = \frac{1}{1+(\omega/\omega_c)^{2N}} \end{aligned}$$</div><p></p>
<p>Butterworth&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x9636;&#x6570;&#x9009;&#x62E9;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; |H_a(j\omega_p)|^2 = \frac{1}{1+(\omega_p/\omega_c)^{2N}} = \frac{1}{\epsilon^2 + 1} \\ &amp; |H_a(j\omega_s)|^2 = \frac{1}{1+(\omega_s/\omega_c)^{2N}} = \frac{1}{A^2} \\ &amp; N = \frac{1}{2} \frac{\log_{10}[(A^2-1)/\epsilon^2]}{\log_{10}(\omega_s/\omega_p)} = \frac{\log_{10}(1/k_1)}{\log_{10}(1/k)} \end{aligned}$$</div><p></p>
<h3 class="mume-header" id="chebyshev%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">Chebyshev&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;</a></h3>

<h4 class="mume-header" id="chebyshev-i%E5%9E%8B%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">Chebyshev I&#x578B;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;</a></h4>

<p>Chebyshev I&#x578B;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x5E45;&#x9891;&#x54CD;&#x5E94;&#x548C;&#x8F6C;&#x79FB;&#x51FD;&#x6570;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; |H_a(j\omega)|^2 = \frac{1}{1+\epsilon^2T_N^2(\omega/\omega_p)} \\ &amp; T_N(\omega)= \begin{cases} \cos(N\cos^{-1}(\omega)), |\omega| \le 1 \\ \cosh(N\cosh^{-1}{\omega}), |\omega| \gt 1 \end{cases} \\ &amp; \epsilon = \sqrt{\frac{2\delta_p}{1-\delta_p}} \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H_a(s) = \frac{z(s)}{p(s)} = \frac{\omega_c^N}{\prod_{l=1}^{N}(s-p_l)}, p_l=\delta_l+j\omega_l \\ &amp; \delta_l = -\omega_p\xi\sin[\frac{(2l--1)\pi}{2N}], \omega_l=\omega_p\zeta\cos[\frac{(2l-1)\pi}{2N}] \\ &amp; \xi=\frac{\gamma^2-1}{2\gamma}, \zeta=\frac{\gamma^2+1}{2\gamma}, \gamma=(\frac{1+\sqrt{1+\epsilon^2}}{\epsilon})^{1/N} \end{aligned}$$</div><p></p>
<p>Chebyshev I&#x578B;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x9636;&#x6570;&#x9009;&#x62E9;:</p>
<p></p><div class="mathjax-exps">$$N = \frac{\cosh^{-1}(1/k_1)}{\cosh^{-1}(1/k)}$$</div><p></p>
<h4 class="mume-header" id="chebyshev-ii%E5%9E%8B%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">Chebyshev II&#x578B;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;</a></h4>

<p>Chebyshev II&#x578B;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x5E45;&#x9891;&#x54CD;&#x5E94;&#x548C;&#x8F6C;&#x79FB;&#x51FD;&#x6570;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; |H_a(j\omega)|^2 = \frac{1}{1+\epsilon^2[\frac{T_N(\omega_s/\omega_p)}{T_N(\omega_s/\omega)}]^2} \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H_a(s) = \frac{z(s)}{p(s)} = \frac{z_l}{\prod_{l=1}^{N}(s-p_l)}, p_l=\delta_l+j\omega_l \\ \end{aligned}$$</div><p></p>
<p>Chebyshev II&#x578B;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x9636;&#x6570;&#x9009;&#x62E9;:</p>
<p></p><div class="mathjax-exps">$$N = \frac{\cosh^{-1}(1/k_1)}{\cosh^{-1}(1/k)}$$</div><p></p>
<h3 class="mume-header" id="%E6%A4%AD%E5%9C%86%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">&#x692D;&#x5706;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;</a></h3>

<p>&#x692D;&#x5706;&#x8FD1;&#x4F3C;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x5E45;&#x9891;&#x54CD;&#x5E94;&#x548C;&#x9636;&#x6570;&#x9009;&#x62E9;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; |H_a(j\omega)|^2 = \frac{1}{1+\epsilon^2R_N^2(\omega/\omega_p)} \\ &amp; N \cong \frac{2\log_{10}(4/k_1)}{\log_{10}(1/\rho)} \\ &amp; \begin{cases} k&apos; = \sqrt{1-k^2} \\ \rho_0 = \frac{1-\sqrt{k&apos;}}{2(1+\sqrt(k&apos;))} \\ \rho = \rho_0 + 2(\rho_0)^5 + 15(\rho_0)^9 + 150(\rho_0)^{13} \end{cases} \end{aligned}$$</div><p></p>
<h3 class="mume-header" id="bessel%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">Bessel&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;</a></h3>

<p>Bessel&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x8F6C;&#x79FB;&#x51FD;&#x6570;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H_a(s) = \frac{d_0}{B_N(s)}=\frac{d_0}{s^N+d_{N-1}s^{N-1}+\cdots+d_1s+d0} \\ &amp; B_N(s) = (2N-1)B_{N-1}(s) + s^2B_{N-2}(s) \\ &amp; B_1(s) = s+1 \\ &amp; B_2(s) = s^2+3s+3 \\ &amp; d_l = \frac{(2N-l)!}{2^{N-l}l!(N-l)!} \end{aligned}$$</div><p></p>
<h2 class="mume-header" id="%E6%A8%A1%E6%8B%9F%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8%E7%9A%84%E8%BD%AC%E6%8D%A2toc"><a href="#toc">&#x6A21;&#x62DF;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x8F6C;&#x6362;</a></h2>

<p>&#x8BB0;&#x6709;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;<span class="mathjax-exps">$H_{LP}(s)$</span>.</p>
<h3 class="mume-header" id="%E6%A8%A1%E6%8B%9F%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8%E8%BD%AC%E4%B8%BA%E9%AB%98%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">&#x6A21;&#x62DF;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x8F6C;&#x4E3A;&#x9AD8;&#x901A;&#x6EE4;&#x6CE2;&#x5668;</a></h3>

<p>&#x4ECE;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;<span class="mathjax-exps">$s$</span>&#x6620;&#x5C04;&#x5230;&#x9AD8;&#x901A;&#x6EE4;&#x6CE2;&#x5668;<span class="mathjax-exps">$\hat{s}$</span>, &#x6709;&#x5982;&#x4E0B;&#x5173;&#x7CFB;:</p>
<p></p><div class="mathjax-exps">$$s=\frac{\omega_p \hat{\omega_p}}{\hat{s}}$$</div><p></p>
<h3 class="mume-header" id="%E6%A8%A1%E6%8B%9F%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8%E8%BD%AC%E4%B8%BA%E5%B8%A6%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">&#x6A21;&#x62DF;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x8F6C;&#x4E3A;&#x5E26;&#x901A;&#x6EE4;&#x6CE2;&#x5668;</a></h3>

<p>&#x4ECE;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;<span class="mathjax-exps">$s$</span>&#x6620;&#x5C04;&#x5230;&#x5E26;&#x901A;&#x6EE4;&#x6CE2;&#x5668;<span class="mathjax-exps">$\hat{s}$</span>, <span class="mathjax-exps">$\omega_o$</span>&#x4E3A;&#x5E26;&#x901A;&#x4E2D;&#x5FC3;&#x70B9;, <span class="mathjax-exps">$\omega_{p2},\omega_{p1}$</span>&#x4E3A;&#x5E26;&#x901A;&#x7684;&#x8FB9;&#x7F18;&#x9891;&#x7387;, &#x6709;&#x5982;&#x4E0B;&#x5173;&#x7CFB;:</p>
<p></p><div class="mathjax-exps">$$s=\omega_p\frac{\hat{s^2}+\hat{\omega_o^2}}{\hat{s}(\hat{\omega_{p2} - \hat{\omega_{p1}}})}$$</div><p></p>
<h3 class="mume-header" id="%E6%A8%A1%E6%8B%9F%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8%E8%BD%AC%E4%B8%BA%E5%B8%A6%E9%98%BB%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">&#x6A21;&#x62DF;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x8F6C;&#x4E3A;&#x5E26;&#x963B;&#x6EE4;&#x6CE2;&#x5668;</a></h3>

<p>&#x4ECE;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;<span class="mathjax-exps">$s$</span>&#x6620;&#x5C04;&#x5230;&#x5E26;&#x963B;&#x6EE4;&#x6CE2;&#x5668;<span class="mathjax-exps">$\hat{s}$</span>, <span class="mathjax-exps">$\omega_o$</span>&#x4E3A;&#x963B;&#x5E26;&#x4E2D;&#x5FC3;&#x70B9;, <span class="mathjax-exps">$\omega_{s2},\omega_{s1}$</span>&#x4E3A;&#x963B;&#x5E26;&#x7684;&#x7ED3;&#x675F;&#x548C;&#x8D77;&#x59CB;&#x9891;&#x7387;, &#x6709;&#x5982;&#x4E0B;&#x5173;&#x7CFB;:</p>
<p></p><div class="mathjax-exps">$$s = \omega_s\frac{\hat{s}(\hat{\omega_{s2}} - \hat{\omega_{s1}})}{\hat{\omega_s^2} + \hat{\omega_o^2}}$$</div><p></p>
<h2 class="mume-header" id="%E6%95%B0%E5%AD%97%E6%BB%A4%E6%B3%A2%E5%99%A8%E8%AE%BE%E8%AE%A1toc"><a href="#toc">&#x6570;&#x5B57;&#x6EE4;&#x6CE2;&#x5668;&#x8BBE;&#x8BA1;</a></h2>

<h3 class="mume-header" id="iir%E6%95%B0%E5%AD%97%E6%BB%A4%E6%B3%A2%E5%99%A8%E8%AE%BE%E8%AE%A1toc"><a href="#toc">IIR&#x6570;&#x5B57;&#x6EE4;&#x6CE2;&#x5668;&#x8BBE;&#x8BA1;</a></h3>

<p>&#x89C1;<a href="%E4%BF%A1%E5%8F%B7%E4%B8%8E%E7%BA%BF%E6%80%A7%E7%B3%BB%E7%BB%9F.md#%E4%BB%8E%E6%8B%89%E6%B0%8F%E5%8F%98%E6%8D%A2%E5%88%B0z%E5%8F%98%E6%8D%A2$%5Cmathscr%7BL%7D%5Cto%5Cmathscr%7Bz%7D$">&#x4FE1;&#x53F7;&#x4E0E;&#x7EBF;&#x6027;&#x7CFB;&#x7EDF;.md</a>, &#x6709;:</p>
<p></p><div class="mathjax-exps">$$s=\frac{1}{T}\ln{z} \approx \frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}})$$</div><p></p>
<p>&#x6545;&#x4ECE;s&#x57DF;&#x7684;&#x865A;&#x8F74;&#x6620;&#x5C04;&#x5230;z&#x57DF;&#x7684;&#x5355;&#x4F4D;&#x5706;&#x4E0A;(&#x6536;&#x655B;&#x7684;&#x4E34;&#x754C;&#x6761;&#x4EF6;)&#x6709;:</p>
<p></p><div class="mathjax-exps">$$j\Omega = \frac{2}{T}\frac{1-e^{-j\omega}}{1+e^{j\omega}} = j\frac{2}{T}\tan(\frac{\omega}{2}) \Rightarrow \Omega=\frac{2}{T}\tan(\frac{\omega}{2})$$</div><p></p>
<p>&#x56E0;&#x6B64;, IIR&#x6570;&#x5B57;&#x6EE4;&#x6CE2;&#x5668;&#x8BBE;&#x8BA1;&#x6B65;&#x9AA4;:</p>
<ol>
<li>&#x786E;&#x5B9A;&#x6EE4;&#x6CE2;&#x5668;&#x53C2;&#x6570;: &#x901A;&#x5E26;&#x8FB9;&#x7F18;&#x9891;&#x7387;, &#x963B;&#x5E26;&#x8D77;&#x59CB;&#x9891;&#x7387;, &#x901A;&#x5E26;&#x6CE2;&#x7EB9;, &#x963B;&#x5E26;&#x6CE2;&#x7EB9;, &#x9009;&#x62E9;&#x53C2;&#x6570;&#x548C;&#x8FA8;&#x522B;&#x53C2;&#x6570;;</li>
<li>&#x6839;&#x636E;&#x6EE4;&#x6CE2;&#x5668;&#x53C2;&#x6570;&#x9009;&#x62E9;&#x6A21;&#x62DF;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7C7B;&#x578B;, &#x5E76;&#x786E;&#x5B9A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x9636;&#x6570;, &#x5F97;&#x5230;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x8F6C;&#x79FB;&#x51FD;&#x6570;;</li>
<li>&#x518D;&#x5C06;&#x6240;&#x5F97;&#x5230;&#x7684;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x8F6C;&#x4E49;&#x51FD;&#x6570;&#x91CD;&#x65B0;&#x6620;&#x5C04;&#x5230;&#x6240;&#x9700;&#x8981;&#x7684;&#x9AD8;&#x901A;/&#x5E26;&#x901A;&#x7B49;&#x8F6C;&#x79FB;&#x51FD;&#x6570;;</li>
<li>&#x5C06;&#x516C;&#x5F0F;<span class="mathjax-exps">$s=\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}})$</span>&#x5E26;&#x5165;&#x5230;&#x6A21;&#x62DF;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x8F6C;&#x79FB;&#x51FD;&#x6570;, &#x4E8E;&#x662F;&#x5F97;&#x5230;z&#x57DF;&#x7684;&#x8F6C;&#x79FB;&#x51FD;&#x6570;;</li>
</ol>
<h4 class="mume-header" id="iir%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8%E8%BD%AC%E6%8D%A2toc"><a href="#toc">IIR&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x8F6C;&#x6362;</a></h4>

<p>&#x6709;&#x65F6;&#x4F1A;&#x9047;&#x5230;&#x8FD9;&#x6837;&#x7684;&#x9700;&#x6C42;: &#x5DF2;&#x7ECF;&#x8BBE;&#x8BA1;&#x597D;&#x4E86;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x7ED3;&#x6784;, &#x4F46;&#x662F;&#x9700;&#x8981;&#x6539;&#x53D8;&#x8BE5;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x7684;&#x622A;&#x6B62;&#x9891;&#x7387;. &#x6B64;&#x65F6;, &#x4E00;&#x822C;&#x6CA1;&#x6709;&#x5FC5;&#x8981;&#x91CD;&#x65B0;&#x8BBE;&#x8BA1;&#x6EE4;&#x6CE2;&#x5668;&#x7ED3;&#x6784;, &#x53EF;&#x4EE5;&#x518D;&#x5F53;&#x524D;&#x6EE4;&#x6CE2;&#x5668;&#x4E0A;&#x518D;&#x4E58;&#x4EE5;&#x4E00;&#x4E2A;&#x5168;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x5373;&#x53EF;&#x5B8C;&#x6210;&#x622A;&#x6B62;&#x9891;&#x7387;&#x7684;&#x8F6C;&#x6362;. &#x8BB0;&#x9700;&#x4ECE;&#x622A;&#x6B62;&#x9891;&#x7387;&#x4E3A;<span class="mathjax-exps">$\omega_c$</span>&#x7684;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x53D8;&#x6362;&#x5230;<span class="mathjax-exps">$\omega_c&apos;$</span>&#x7684;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; \because |H_{ap}(e^{j\omega})|^2 = 1 \\ &amp; \therefore H_{ap}(e^{j\omega}) = \pm\frac{d_N+d_{N-1}e^{-j\omega}+\cdots+d_1e^{-j\omega(N-1)}+e^{-j\omega N}}{1+d_1e^{-j\omega}+\cdots+d_{N-1}e^{-j\omega(N-1)} +d_Ne^{-j\omega N}} \Rightarrow \\ &amp; \quad H_{ap}(z) = \pm \frac{d_N+d_{N-1}z^{-1} + \cdots+d_1z^{-N+1}+z^{-N}}{1+d_1z^{-N}+\cdots+d_{N-1}z^{1-N}+d_Nz^{-N}} = \pm \prod_{i=1}^{N}\frac{-\lambda_i^* + z^{-1}}{1-\lambda_iz^{-1}} \\ &amp; \quad H(z&apos;)=H(H_{ap}(z)) \Rightarrow H_{ap}(z) = \frac{1-\lambda z&apos;}{z&apos; - \lambda} \Rightarrow \\ &amp; \quad e^{-j\omega}=\frac{e^{-j\omega&apos;} - \lambda}{1-\lambda e^{-j\omega&apos;}} \Rightarrow \tan(\omega_c/2)=\frac{1+\lambda}{1-\lambda}\tan(\omega_c&apos;/2) \Rightarrow \\ &amp; \quad \lambda = \frac{\sin(\frac{\omega_c-\omega_c&apos;}{2})}{\sin(\frac{\omega_c-\omega_c&apos;}{2})} \end{aligned}$$</div><p></p>
<h3 class="mume-header" id="fir%E6%95%B0%E5%AD%97%E6%BB%A4%E6%B3%A2%E5%99%A8%E8%AE%BE%E8%AE%A1toc"><a href="#toc">FIR&#x6570;&#x5B57;&#x6EE4;&#x6CE2;&#x5668;&#x8BBE;&#x8BA1;</a></h3>

<p>FIR&#x4E00;&#x822C;&#x7528;&#x6765;&#x8BBE;&#x8BA1;&#x7EBF;&#x6027;&#x76F8;&#x4F4D;&#x6570;&#x5B57;&#x6EE4;&#x6CE2;&#x5668;, &#x6240;&#x8C13;&#x7EBF;&#x6027;&#x76F8;&#x4F4D;&#x9700;&#x6EE1;&#x8DB3;&#x6052;&#x7FA4;&#x5EF6;&#x65F6;<span class="mathjax-exps">$\frac{\rm{d}\theta(\omega)}{\rm{d}\omega}=-\tau$</span>, &#x6216;&#x8005;&#x6052;&#x7FA4;&#x5EF6;&#x65F6;&#x548C;&#x6052;&#x76F8;&#x5EF6;&#x65F6;<span class="mathjax-exps">$\frac{\theta(\omega)}{\omega}=-\tau$</span>&#x90FD;&#x6EE1;&#x8DB3;. &#x5176;&#x4E2D;, <span class="mathjax-exps">$\tau$</span>&#x4E3A;&#x5E38;&#x6570;.</p>
<p>&#x5F53;&#x6052;&#x76F8;&#x5EF6;&#x65F6;&#x548C;&#x6052;&#x7FA4;&#x5EF6;&#x65F6;&#x540C;&#x65F6;&#x6210;&#x7ACB;&#x65F6;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H(e^{j\omega}) = \sum_{n=0}^{N}h(n)\cdot(\cos(\omega n) - j\sin(\omega n)) \Rightarrow \\ &amp; \tan(\theta(\omega))=\tan(-\tau \omega) \Rightarrow \frac{\sin(\tau \omega)}{\cos(\tau \omega)}=\frac{\sum_{n=0}^{N}h(n)\sin(\omega n)}{\sum_{n=0}^{N}h(n)\cos(\omega n)} \Rightarrow \\ &amp; \sum_{n=0}^{N}h(n)sin(\omega (\tau - n)) = 0 \Rightarrow \\ &amp; \begin{cases} \tau = \frac{N}{2} \\ h(n) = h(N-n), 0 \le n \le N \end{cases} \end{aligned}$$</div><p></p>
<p>&#x5F53;&#x4EC5;&#x6EE1;&#x8DB3;&#x6052;&#x7FA4;&#x5EF6;&#x65F6;&#x65F6;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H(e^{j\omega}) = \sum_{n=0}^{N}h(n)\cdot(\cos(\omega n) - j\sin(\omega n)) \Rightarrow \\ &amp; \tan(\theta(\omega))=\tan(\theta_0 -\tau \omega) \Rightarrow \frac{\sin(-\theta_0 + \tau \omega)}{\cos(-\theta_0 + \tau \omega)}=\frac{\sum_{n=0}^{N}h(n)\sin(\omega n)}{\sum_{n=0}^{N}h(n)\cos(\omega n)} \Rightarrow \\ &amp; \sum_{n=0}^{N}h(n)sin(-\theta_0 + \omega (\tau - n)) = 0 \Rightarrow \\ &amp; \begin{cases} \tau = \frac{N}{2} \\ h(n) = -h(N-n), 0 \le n \le N \end{cases} \end{aligned}$$</div><p></p>
<p>&#x5F53;&#x8F6C;&#x79FB;&#x51FD;&#x6570;&#x6EE1;&#x8DB3;&#x7EBF;&#x6027;&#x76F8;&#x4F4D;&#x65F6;, &#x6709;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H(z) = \pm\sum_{n=0}^{N}h(N-n)z^{-n} = \pm z^{1-N}H(z^{-1}) \\ &amp; \exists\quad &#x5171;&#x8F6D;&#x955C;&#x9762;&#x5BF9;&#x79F0;&#x7684;&#x96F6;&#x70B9;z_i, z_i^*, 1/z_i, 1/z_i^* \end{aligned}$$</div><p></p>
<p>&#x7EFC;&#x4E0A;, &#x6839;&#x636E;&#x91C7;&#x6837;&#x70B9;&#x5947;&#x5076;&#x6570;&#x4E0D;&#x540C;, &#x53CA;&#x7EBF;&#x6027;&#x76F8;&#x4F4D;&#x6EE1;&#x8DB3;&#x6761;&#x4EF6;&#x4E0D;&#x540C;, &#x6709;4&#x4E2D;&#x7EBF;&#x6027;&#x76F8;&#x4F4D;FIR&#x6EE4;&#x6CE2;&#x5668;.</p>
<h4 class="mume-header" id="hn%E5%81%B6%E5%AF%B9%E7%A7%B0%E4%B8%94%E5%81%B6%E6%95%B0%E4%B8%AA%E9%87%87%E6%A0%B7%E7%82%B9toc"><a href="#toc">h(n)&#x5076;&#x5BF9;&#x79F0;&#x4E14;&#x5076;&#x6570;&#x4E2A;&#x91C7;&#x6837;&#x70B9;</a></h4>

<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H(z) = \sum_{n=0}^{(N-1)/2}h(n)\cdot(z^{-n}+z^{-(N-n)}) \Rightarrow \\ &amp; \begin{cases} |H(\omega)| = 2\sum_{n=1}^{\frac{N+1}{2}}h(\frac{N+1}{2}-n)\cos((n-\frac{1}{2})\omega) \\ \theta(\omega) = -\omega (\frac{N}{2}) \end{cases} \end{aligned}$$</div><p></p>
<p><span class="mathjax-exps">$\cos((n-\frac{1}{2})\omega)$</span>&#x5728;<span class="mathjax-exps">$\omega=\pi$</span>&#x65F6;&#x4E3A;0, &#x56E0;&#x6B64;&#x4E0D;&#x9002;&#x5408;&#x7528;&#x6765;&#x8BBE;&#x8BA1;&#x9AD8;&#x901A;/&#x5E26;&#x963B;&#x6EE4;&#x6CE2;&#x5668;(&#x5728;<span class="mathjax-exps">$\omega=\pi$</span>&#x5904;&#x4E0D;&#x4E3A;0), &#x53EF;&#x7528;&#x6765;&#x8BBE;&#x8BA1;&#x4F4E;&#x901A;/&#x5E26;&#x901A;FIR&#x6570;&#x5B57;&#x6EE4;&#x6CE2;&#x5668;.</p>
<h4 class="mume-header" id="hn%E5%81%B6%E5%AF%B9%E7%A7%B0%E4%B8%94%E5%A5%87%E6%95%B0%E4%B8%AA%E9%87%87%E6%A0%B7%E7%82%B9toc"><a href="#toc">h(n)&#x5076;&#x5BF9;&#x79F0;&#x4E14;&#x5947;&#x6570;&#x4E2A;&#x91C7;&#x6837;&#x70B9;</a></h4>

<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H(z) = h(\frac{N}{2}) + \sum_{n=0}^{\frac{N}{2}-1}h(n)\cdot(z^{-n}+z^{-(N-n)}) \Rightarrow \\ &amp; \begin{cases} |H(\omega)| = h(\frac{N}{2}) + 2\sum_{n=1}^{\frac{N}{2}}h(\frac{N}{2}-n)\cos(n\omega) \\ \theta(\omega) = -\omega (\frac{N}{2}) \end{cases} \end{aligned}$$</div><p></p>
<p><span class="mathjax-exps">$\cos(n\omega)$</span>&#x5173;&#x4E8E;<span class="mathjax-exps">$0,\pi,2\pi$</span>&#x5076;&#x5BF9;&#x79F0;, &#x6545;&#x53EF;&#x7528;&#x6765;&#x8BBE;&#x8BA1;&#x4F4E;&#x901A;/&#x9AD8;&#x901A;/&#x5E26;&#x901A;/&#x5E26;&#x963B;FIR&#x6570;&#x5B57;&#x6EE4;&#x6CE2;&#x5668;.</p>
<h4 class="mume-header" id="hn%E5%A5%87%E5%AF%B9%E7%A7%B0%E4%B8%94%E5%81%B6%E6%95%B0%E4%B8%AA%E9%87%87%E6%A0%B7%E7%82%B9toc"><a href="#toc">h(n)&#x5947;&#x5BF9;&#x79F0;&#x4E14;&#x5076;&#x6570;&#x4E2A;&#x91C7;&#x6837;&#x70B9;</a></h4>

<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H(z) = \sum_{n=0}^{(N-1)/2}h(n)\cdot(z^{-n}-z^{-(N-n)}) \\ &amp; \begin{cases} |H(\omega)| = 2\sum_{n=1}^{\frac{N+1}{2}}h(\frac{N+1}{2}-n)\sin((n-\frac{1}{2})\omega) \\ \theta(\omega) =\frac{\pi}{2} -\omega (\frac{N}{2}) \end{cases} \end{aligned}$$</div><p></p>
<p><span class="mathjax-exps">$\sin((n-\frac{1}{2})\omega)$</span>&#x5728;<span class="mathjax-exps">$\omega=0$</span>&#x65F6;&#x4E3A;0, &#x6545;&#x4E0D;&#x53EF;&#x7528;&#x6765;&#x8BBE;&#x8BA1;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;&#x548C;&#x5E26;&#x963B;&#x6EE4;&#x6CE2;&#x5668;, &#x53EF;&#x7528;&#x6765;&#x8BBE;&#x8BA1;&#x9AD8;&#x901A;/&#x5E26;&#x901A;FIR&#x6570;&#x5B57;&#x6EE4;&#x6CE2;&#x5668;.</p>
<h4 class="mume-header" id="hn%E5%A5%87%E5%AF%B9%E7%A7%B0%E4%B8%94%E5%A5%87%E6%95%B0%E4%B8%AA%E9%87%87%E6%A0%B7%E7%82%B9toc"><a href="#toc">h(n)&#x5947;&#x5BF9;&#x79F0;&#x4E14;&#x5947;&#x6570;&#x4E2A;&#x91C7;&#x6837;&#x70B9;</a></h4>

<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; H(z) = \sum_{n=0}^{\frac{N}{2}-1}h(n)\cdot(z^{-n}-z^{-(N-n)}) \\ &amp; \begin{cases} |H(\omega)| = 2\sum_{n=1}^{\frac{N}{2}}h(\frac{N}{2}-n)\sin(n\omega) \\ \theta(\omega) =\frac{\pi}{2} -\omega (\frac{N}{2}) \end{cases} \end{aligned}$$</div><p></p>
<p><span class="mathjax-exps">$\sin(n\omega)$</span>&#x5728;<span class="mathjax-exps">$\omega=0,\pi$</span>&#x5904;&#x4E3A;0, &#x6545;&#x4E0D;&#x53EF;&#x7528;&#x6765;&#x8BBE;&#x8BA1;&#x5E26;&#x963B;/&#x9AD8;&#x901A;/&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;, &#x53EF;&#x7528;&#x6765;&#x8BBE;&#x8BA1;&#x5E26;&#x901A;FIR&#x6570;&#x5B57;&#x6EE4;&#x6CE2;&#x5668;.</p>
<h4 class="mume-header" id="%E7%AA%97%E5%87%BD%E6%95%B0%E6%B3%95%E8%AE%BE%E8%AE%A1fir%E6%95%B0%E5%AD%97%E4%BD%8E%E9%80%9A%E6%BB%A4%E6%B3%A2%E5%99%A8toc"><a href="#toc">&#x7A97;&#x51FD;&#x6570;&#x6CD5;&#x8BBE;&#x8BA1;FIR&#x6570;&#x5B57;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;</a></h4>

<p>&#x7406;&#x60F3;&#x7684;&#x4F4E;&#x901A;&#x6EE4;&#x6CE2;&#x5668;, &#x5176;&#x65F6;&#x57DF;&#x51B2;&#x51FB;&#x54CD;&#x5E94;&#x5E8F;&#x5217;&#x4E3A;<span class="mathjax-exps">$h(n)=\frac{\sin(\omega_c n)}{\pi n}, -\infty\le n \le\infty$</span>, &#x8BE5;&#x5E8F;&#x5217;&#x5E76;&#x4E0D;&#x7EDD;&#x5BF9;&#x6536;&#x655B;, &#x4F46;&#x662F;&#x80FD;&#x91CF;&#x4E3B;&#x8981;&#x96C6;&#x4E2D;&#x5728;&#x4E3B;&#x74E3;&#x4E0A;. &#x56E0;&#x6B64;&#x5B9E;&#x9645;&#x5E94;&#x7528;&#x4E2D;&#x5E38;&#x5E38;&#x52A0;&#x7A97;&#x51FD;&#x6570;&#x622A;&#x65AD;&#x4E3B;&#x74E3;&#x6216;&#x52A0;&#x4E0A;&#x524D;&#x51E0;&#x4E2A;&#x65C1;&#x74E3;<span class="mathjax-exps">$h(n)*W(n)$</span>&#x4F5C;&#x4E3A;&#x7CFB;&#x7EDF;&#x7684;&#x51B2;&#x51FB;&#x54CD;&#x5E94;&#x5E8F;&#x5217;.</p>
<p>&#x5047;&#x8BBE;&#x6240;&#x9700;&#x8981;&#x7684;&#x62BD;&#x6837;&#x70B9;&#x6570;&#x4E3A;<span class="mathjax-exps">$N$</span>, &#x5219;&#x7A97;&#x51FD;&#x6570;&#x7684;&#x7A97;&#x53E3;&#x5927;&#x5C0F;&#x81F3;&#x5C11;&#x4E3A;<span class="mathjax-exps">$2N+1$</span>, &#x5E38;&#x7528;&#x7A97;&#x51FD;&#x6570;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; Rectangular(n) =\begin{cases} 1, -N \le n \le N \\ 0, \quad others \end{cases} \\ &amp; Bartlett(n) = \begin{cases} 1 - \frac{|n|}{N+1}, -N \le n \le N \\ 0, \quad others \end{cases} \\ &amp; Hann(n) = \begin{cases} \frac{1}{2}[1-\cos(\frac{2\pi n}{2N+1})], -N \le n \le N \\ 0, \quad others \end{cases} \\ &amp; Hanmming(n) = \begin{cases} 0.54 - 0.46\cos(\frac{2\pi n}{2N+1}), -N \le n \le N \\ 0, \quad others \end{cases} \\ &amp; Blackman(n) = \begin{cases} 0.42 - 0.5\cos(\frac{2\pi n}{2N+1}) + 0.08\cos(\frac{4\pi n}{2N+1}) \end{cases} \end{aligned}$$</div><p></p>
<p>&#x4EE5;&#x4E0A;&#x51E0;&#x79CD;&#x7A97;&#x51FD;&#x6570;&#x7684;&#x5C5E;&#x6027;&#x5982;&#x4E0B;:</p>
<table>
<thead>
<tr>
<th style="text-align:center">&#x7A97;&#x7C7B;&#x578B;</th>
<th style="text-align:center">&#x4E3B;&#x74E3;&#x5BBD;&#x5EA6;</th>
<th style="text-align:center">&#x65C1;&#x74E3;&#x5CF0;&#x503C;&#x8870;&#x51CF;(dB)</th>
<th style="text-align:center">&#x6700;&#x5C0F;&#x963B;&#x5E26;&#x8870;&#x51CF;(dB)</th>
<th style="text-align:center">&#x8FC7;&#x6E21;&#x5E26;&#x5BBD;&#x5EA6;(<span class="mathjax-exps">$\omega_p-\omega_s$</span>)</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">Rectangular</td>
<td style="text-align:center"><span class="mathjax-exps">$4\pi/(2N+1)$</span></td>
<td style="text-align:center">13.3</td>
<td style="text-align:center">20.9</td>
<td style="text-align:center">0.92<span class="mathjax-exps">$\pi/N$</span></td>
</tr>
<tr>
<td style="text-align:center">Barlett</td>
<td style="text-align:center"><span class="mathjax-exps">$4\pi/(2N+1)$</span></td>
<td style="text-align:center">26.5</td>
<td style="text-align:center">25</td>
<td style="text-align:center"><span class="mathjax-exps">$2.1\pi/N$</span></td>
</tr>
<tr>
<td style="text-align:center">Hann</td>
<td style="text-align:center"><span class="mathjax-exps">$8\pi/(2N+1)$</span></td>
<td style="text-align:center">31.5</td>
<td style="text-align:center">43.9</td>
<td style="text-align:center"><span class="mathjax-exps">$3.11\pi/N$</span></td>
</tr>
<tr>
<td style="text-align:center">Hanmming</td>
<td style="text-align:center"><span class="mathjax-exps">$8\pi/(2N+1)$</span></td>
<td style="text-align:center">42.7</td>
<td style="text-align:center">54.5</td>
<td style="text-align:center"><span class="mathjax-exps">$3.32\pi/N$</span></td>
</tr>
<tr>
<td style="text-align:center">Blackman</td>
<td style="text-align:center"><span class="mathjax-exps">$12\pi/(2N+1)$</span></td>
<td style="text-align:center">58.1</td>
<td style="text-align:center">75.3</td>
<td style="text-align:center"><span class="mathjax-exps">$5.56\pi/N$</span></td>
</tr>
</tbody>
</table>
<p>&#x7EFC;&#x4E0A;, &#x8BBE;&#x8BA1;&#x7B56;&#x7565;: &#x901A;&#x8FC7;&#x8FC7;&#x6E21;&#x5E26;&#x5BBD;&#x5EA6;&#x6C42;&#x51FA;&#x6EE4;&#x6CE2;&#x5668;&#x9636;&#x6570;<span class="mathjax-exps">$N$</span>, &#x6839;&#x636E;&#x6EE4;&#x6CE2;&#x5668;&#x7C7B;&#x578B;/&#x76F8;&#x4F4D;&#x8981;&#x6C42;&#x9009;&#x62E9;FIR&#x7C7B;&#x578B;, &#x6C42;<span class="mathjax-exps">$h(n)\cdot W(n)$</span>&#x5F97;&#x5230;&#x5E8F;&#x5217;<span class="mathjax-exps">$h_w(n)$</span>, &#x5C06;<span class="mathjax-exps">$h_w(n)$</span>&#x5E26;&#x5165;&#x5230;&#x6240;&#x9009;&#x62E9;&#x7684;FIR&#x7C7B;&#x578B;&#x7684;&#x8F6C;&#x79FB;&#x51FD;&#x6570;&#x4E2D;&#x5F97;&#x5230;&#x5E45;&#x9891;&#x54CD;&#x5E94;.</p>
<h2 class="mume-header" id="%E5%8F%82%E8%80%83%E8%B5%84%E6%96%99">&#x53C2;&#x8003;&#x8D44;&#x6599;</h2>

<ol>
<li><a href="https://book.douban.com/subject/7058940/">Digtal Signal Processing, Sanjit K.Mitra</a>.</li>
</ol>

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